Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| automorphism | ? |
| homology | Abelianization (H1) = C25×C4 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, socle, cosocle | C23 | C26 |
| commutator | C22 | C25×C4 |
| prime radical | C2×#3531 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 36 points:
(5,6)(7,8)(9,13)(10,14)(11,15)(12,18)(16,17)(19,20)(21,22)(23,24)(25,29)(26,27)(28,32)(30,31)(33,34)(35,36);(5,7)(6,8)(9,10)(11,12)(13,14)(15,18)(16,19)(17,20)(22,27)(23,28)(29,34)(30,35);(5,8)(6,12)(7,15)(9,17)(10,13)(11,18)(14,19)(16,20)(21,23,26,28)(22,24,27,32)(25,30,33,35)(29,31,34,36);(5,9)(6,13)(7,16)(8,17)(10,12)(11,19)(14,18)(15,20)(21,24)(22,28)(23,27)(25,31)(26,32)(29,35)(30,34)(33,36);(1,2,3,4)(5,8)(6,7)(9,17)(10,20)(11,18)(12,15)(13,16)(14,19);(5,10,11,16)(6,14,15,17)(7,9,12,19)(8,13,18,20)(21,25,26,33)(22,29,27,34)(23,30,28,35)(24,31,32,36)Polycyclic presentation:
2^9|c^2=g,e^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh140 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7534606);